Fekete-like polynomials

FEKETE-LIKE POLYNOMIALS

KEVIN G. HARE AND SOROOSH YAZDANI

Dedicated to Peter Borwein on the occasion of his 55th birthday.

Abstract. In 2001, Borwein, Choi, and Yazdani looked at an extremal prop-erty of a class of polynomial with ±1 coefficients. Their key result was:

Theorem (Borwein, Choi, Yazdani, 2001). Let f(z) = ±z ± z2 ± · · · ± zN−1,

and ζ a primitive Nth root of unity. If N is an odd positive integer then

maxi

|f(ζi)| ≥√

N

with equality if and only if N is an odd prime.

Moreover, if equality holds, they gave an explicit construction for f(z). Inthis paper, we look at the case when N is even. In particular, we investigatethe following

Conjecture. Let f(z) and ζ be as above. If N > 2 is an even positive integer

then

maxi

|f(ζi)| ≥√

N + 1

with equality if and only if N + 1 is a power of an odd prime.

This conjecture was made after extensive computations. Partial resultstowards proving this conjecture are given.

1. Introduction

Recall that the set of Littlewood polynomials is defined as ±zN±zN−1±· · ·±z±1.For technical reasons, we find it more convenient to consider the set of polynomialswith constant term 0, that is z times a Littlewood polynomial. Define the set ofLittlewood-like polynomials of degree N − 1 as

LN ={±z ± z2 ± · · · ± zN−1

}.

For f =∑N−1

i=1 aizi ∈ LN , let f∗ =

∑N−1i=1 aN−iz

i be the reciprocal polynomial (i.e.f∗(z) = zNf(1/z)), and

AN ={

f(z) ∈ LN | f(z) = (−1)N/2f∗(−z)}

be the subset of anti-skewsymmetric polynomials. We let a0 = 0 throughout thispaper. Note that if f(z) =

∑i aiz

i ∈ AN then aN−i = (−1)N/2+iai. We say that apolynomial f ∈ LN or AN that satisfies equation (1.1) with equality is an optimal

polynomial.In [5], the case when N is odd was investigated:

Date: January 27, 2010.2000 Mathematics Subject Classification. Primary: 11J54, 11B83, 12-04.Research of K. G. Hare supported, in part by NSERC of Canada.Research of S. Yazdani supported, in part by NSERC of Canada .The authors would like to thank IRMACS (SFU) for their hospitality and support.

1

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2 KEVIN G. HARE AND SOROOSH YAZDANI

Theorem 1.1 (Borwein, Choi, Yazdani, 2001). Let f(z) =∑N−1

i=1 aizi ∈ LN and

ζ a primitive N th root of unity. If N is an odd positive integer then

maxi

|f(ζi)| ≥√

N

with equality if and only if N is prime. Moreover, if equality holds, then a1f(z) is

the Fekete polynomial, that is ai = a1

(iN

), where

(·N

)is the Legendre symbol.

Notice, if N is a prime number, and f is the Fekete polynomial of degree N − 1,then for ζ a primitive Nth root of unity we have that |f(ζi)| =

√N for i =

1, 2, · · · , N − 1 and f(1) = 0.An obvious question that arose from this study was: what happens if N is even?Based on extensive calculation, the authors make the following

Conjecture 1.2. Let f(z) = ±z ± z2 ± · · · ± zN−1. Let ζ be a primitive N th root

of unity. If N > 2 is an even positive integer then

(1.1) maxi

|f(ζi)| ≥√

N + 1

with equality if and only if N + 1 is a power of an odd prime.

Specifically this conjecture is verified for N ≤ 42 for all Littlewood-like polyno-mials, and for N ≤ 84 for anti-skewsymmetric polynomials. More on these, andother computations can be found in Section 5.

Throughout this paper, let p be an odd prime and q = pk a prime power. LetFq be the finite field with q elements in it. We denote F∗

q the group of invertibleelements in Fq. Let χ : Fq → {0,±1} be the quadratic residue map, that is χ(c)+1is the number of solutions to z2 = c in Fq. It is well known that χ restricted to F∗

q

is a group homomorphism to the group of two elements. If N + 1 = q = pk, thenFq will have a primitive Nth root of unity, which we will denote by r.

Note that LN ⊂ Z[x], however in some cases it is useful to treat LN ⊂ Fq[x].When N + 1 = q, for any choice of ζ and r a primitive Nth root of unity, there isa natural map π taking Z[ζ] to Fq, sending ζ to r.

In Section 2 for N even, we show that:

Theorem 1.3. Let f ∈ AN , and ζ a primitive N th root of unity. If N > 2 is an

even number then

(1.2) maxi

|f(ζi)| ≥√

N + 1.

Furthermore, if equality is achieved then |f(ζi)| =√

N + 1 for all but two values of

i ∈ {0, 1, . . . , N − 1}. At these other two values of i, we have |f(ζi)| = 1.

In the above theorem, we needed the assumption that f ∈ AN in the proof. Thisis somewhat unfortunate, because computationally it appears that for f ∈ LN theinequality 1.2 is always satisfied, and when it is an equality we have f ∈ AN (seeSection 5). If we could replace AN in Theorem 1.3 with LN , then we would haveproved one part of Conjecture 1.2. The proof is similar to the case when N is odd,although we have to work slightly more because the parity argument in [5] fails inthis case.

In Section 3, for N +1 a prime power, we construct a Littlewood-like polynomialg satisfying maxi |g(ζi)| =

√N + 1. Most f found satisfy this property, (see Section

5), but there are a few unusual exceptions to this rule. None of these exceptions

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FEKETE-LIKE POLYNOMIALS 3

contradict Conjecture 1.2, and none of them occur when N + 1 is not a power ofan odd prime.

We will present some evidence for Conjecture 1.2 in Section 4 by proving it underextra (unfortunately fairly restrictive) assumptions on f .

In Section 5 we give some computational evidence in support of our conjecture.In addition, we make some concluding comments, and list some possible futuredirections for this work.

2. A Proof of Theorem 1.3

The proof of this theorem is similar to the odd N case, although the parityargument needs to be modified. Let ζ ∈ C be a primitive Nth root of unity. Let

h(z) =

N−1∑

k=0

ckzk

where ck =∑

j−ℓ≡k ajaℓ and the sum is over all 0 ≤ j, ℓ < N with j − ℓ ≡ k

(mod N). (Recall that a0 = 0.) We have the following:

N−1∑

i=0

|f(ζi)|2 =

N−1∑

i=0

f(ζi)f(ζ−i)

=

N−1∑

i=0

N−1∑

l=1

N−1∑

j=1

ajalζi(j−l)

=N−1∑

i=0

N−1∑

k=0

∑

j−l≡k

ajalζki

=

N−1∑

i=0

h(ζi)

= Nc0

= N(N − 1).(2.1)


Looking at the 4-norm gives us:

N−1∑

i=0

|f(ζi)|4 =

N−1∑

i=0

(f(ζi)f(ζ−i)

) (f(ζ−i)f(ζi)

)

=N−1∑

i=0

h(ζi)h(ζ−i)

=

N−1∑

i=0

N−1∑

j=0

N−1∑

k=0

cjckζi(j−k)

=

N−1∑

k=0

N−1∑

j=0

cjck

N−1∑

i=0

ζi(j−k)

= N

N−1∑

j=0

c2j

= N(N − 1)2 + NN−1∑

j=1

c2j .(2.2)

Now if ajaℓ 6= 0 we have ajaℓ ≡ aj + aℓ − 1 (mod 4). Furthermore, a0ak = 0 ≡a0 + ak − 1 + (1 − ak) (mod 4). Therefore

ck =∑

j−ℓ≡k

ajaℓ

≡

∑

j−ℓ≡k

aj + aℓ − 1

+ (1 − ak) + (1 − aN−k) (mod 4)

≡ 2

∑

j

aj

− N + 2 − (ak + aN−k) (mod 4)

≡ 2f(1) + N + 2 − (ak + aN−k) (mod 4)

≡ 2 + N + 2 − (ak + aN−k) (mod 4)

≡ N − (ak + aN−k) (mod 4)

Since we are assuming that f ∈ AN , we have that aN−k = (−1)k+N/2ak, we get

ck ≡ N − ak(1 + (−1)k+N/2) (mod 4)

≡ N + (1 + (−1)k+N/2) (mod 4)

≡{

2 if k even,

0 if k odd.(2.3)


Therefore |ck| ≥ 2 when k is even, and so c2k ≥ 4. By equations (2.2) and (2.3) we

haveN−1∑

i=0

|f(ζi)|4 = N(N − 1)2 + N

N−1∑

k=1

c2k

≥ N(N − 1)2 + N

(N − 2

2

)4

= N(N2 − 3).(2.4)

Hence by equation (2.1) we have

N−1∑

i=0

|f(ζi)|2 − 1 = N(N − 2),

which, combined with equation (2.4) gives us

N−1∑

i=0

(|f(ζi)|2 − 1)2 ≥ N2(N − 2).

Note that f is assumed anti-skewsymmetric, and hence

|f(ζi)| = |f(−ζi)| = |f(ζi+N/2)|.Therefore we have

N/2−1∑

i=0

|f(ζi)|2 − 1 = N(N − 2)/2,(2.5)

N/2−1∑

i=0

(|f(ζi)|2 − 1)2 ≥ N2(N − 2)/2.(2.6)

Let xi = N+1−|f(ζi)|2

N . From (2.5) and (2.6) we get that∑

xi = 1∑

x2i ≥ 1

If maxi |f(ζi)| ≤√

N + 1 then 0 ≤ xi, and since∑

xi = 1 we have xi ≤ 1. Hencex2

i ≤ xi, for all i. But this gives that

1 ≤∑

x2i ≤

∑xi = 1

which implies that x2i = xi (hence xi ∈ {0, 1}) for all i. In particular, this implies

that xi = 0 for all except exactly one xi. Translating back to information aboutf(ζ) we get

maxi

|f(ζi)| ≥√

N + 1.

If equality holds, then |f(ζi)| =√

N + 1 for all but two values of i, where the valueat the remaining two values is 1.

By noticing that f ∈ AN we have

|f(ζi)| = |f(ζ−i)| = |f(−ζi)| = |f(−ζ−i)|hence we see that if f(ζi) = ±1, then ζi = ±1, or ζi = ±

√−1.


3. A construction giving equality

In this section, we will give a constructive proof for a polynomial with the desiredproperty, given by (3.2).

Theorem 3.1. Let N = q − 1 = pk − 1 be one less than a prime power. Let r be a

primitive N th root of unity in Fq. Define

g(z) =

N−1∑

i=1

aizi,(3.1)

with ai = χ(ri − 1) where χ is the quadratic residue map. Then g(z) ∈ AN , and

has the desired property that

(3.2) maxi

|g(ζi)| =√

N + 1

where ζ is a primitive N th root of unity.

We say that a polynomial ±g(z) constructed via Theorem 3.1 is a Fekete-like

polynomial. We first need the following well known

Lemma 3.2. For any b ∈ Fq we have

∑

a∈Fq

χ(a)χ(a + b) =

{−1 if b 6= 0,

q − 1 if b = 0.(3.3)

See [7] for discussions on Lemma 3.2, and more general problems relating to it.

Proof of Theorem 3.1. We use Lemma 3.2 to calculate |g(ζk)|2. Note that

|g(ζk)|2 = g(ζk)g(ζ−k)

=∑

ℓ,j

χ(rℓ − 1)χ(rj − 1)ζk(ℓ−j)

=∑

i,j

χ(ri+j − 1)χ(rj − 1)ζki

=∑

i,j

χ(ri)χ(rj − r−i)χ(rj − 1)ζki

=∑

i,j

χ(ri)χ(rj − 1 − (r−i − 1))χ(rj − 1)ζki

=∑

i

(−1)iζki∑

j

χ(rj − 1)χ(rj − 1 − (r−i − 1)).

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Let 1 − r−i = b, then the inner sum becomes

∑

j

χ(rj − 1)χ(rj − 1 + b) =

∑

a∈Fq

χ(a)χ(a + b)

− χ(−1)χ(−1 + b)

=

∑

a∈Fq

χ(a)χ(a + b)

− χ(1 − (1 − r−i))

=

∑

a∈Fq

χ(a)χ(a + b)

− χ(r−i)

=

{−1 − (−1)i if b 6= 0,

q − 2 if b = 0.

Therefore the inner sum is just −1 − (−1)i, and hence

|g(ζk)|2 = q − 2 +

q−2∑

i=1

(−1)iζki(−1 − (−1)i)

= q −q−2∑

i=0

(ζki + (−ζ)ki

)

=

{1 if ζk = ±1,

q otherwise.

Therefore we get that the polynomial g satisfies the desired result. �

4. Uniqueness

In this section we study how easy it is for a Littlewood-like polynomial to satisfyequality of Conjecture 1.2. Specifically, for f ∈ LN (or even f ∈ AN ), if we have

|f(ζk)|2 =

{1 if ζk = ±1,

q otherwise,(4.1)

then what properties does f satisfy?Assume that f satisfies condition (4.1) above. Let p|N+1 and let N |q−1 = pk−1.

(If N + 1 is a power of a prime, then q = N + 1.) Let ζ be a primitive Nth rootof unity in C, and r be a primitive Nth root of unity in Fq. Let π : Z[ζ] → Fq by

π(ζ) = r. Note that for any k, with N2 ∤ k we have that |f(ζk)|2 = f(ζk)f∗(ζk) = q.

Therefore, in this case π(f(ζk)f(ζ−k)) = 0, which implies f(rk)f(r−k) = 0, wheref is the image of f in Fq[z]. For the rest of this section we will focus our attentionto polynomials over Fq, and as such, to simplify notation, we will use f instead

of f . This gives us that {r±1, r±2, · · · , r±(N/2−1)} are all roots of f(z)f∗(z), orequivalently that

xN − 1

x2 − 1

∣∣∣∣ f(z)f∗(z),

where all polynomials are elements of Fq[z]. Computationally it seems that wehave tighter conditions on the roots most of the times. Namely for most optimalpolynomials there exists a primitive Nth root of unity r such that f(ri) = 0 for


i = 1, 2, . . . , N/2 − 1. Of the 700 optimal polynomial found in Section 5, 690 ofthem had this property. (There were 2 for N = 8 that did not, and 8 for N = 58.)See Section 5 for more on the computations.

If we assume that the roots of f ∈ Fq[z] satisfies these conditions we get

Theorem 4.1. Let p|N + 1 and let N |q − 1 = pk − 1. Let r ∈ Fq be a primitive

N th root of unity. Assume that f ∈ LN ⊂ Fq[z] such that f(ri) = 0 for i =1, 2, . . . , N/2 − 1. Then q = N + 1 and f(z) = ±

∑χ(ri − 1)zi.

Proof. Let

f(z) =

N−1∑

i=0

f(r−i)zi.

(This is the Fourier transform of f with respect to r.) If f(z) =∑

i aizi, and

f(z) =∑

i bizi, then

f(r−i) = bi,

f(rj) = Naj

= −aj.

Therefore, by our assumptions we get that f ∈ Fq[z] of degree at most N/2, asbN−i = f(ri) = 0 for i = 1, 2, · · · , N/2 − 1. However, since ai = ±1 we get that

f(1) = 0, and f2(z) = 1 for z any root of (zN − 1)/(z − 1). Therefore

f(z)2 − 1 =zN − 1

z − 1h(z)

for some linear function h(z), since the degree of f is at most N/2 we get that f2

has degree no more than N . Evaluating at 1 we get f(1)2 − 1 = Nh(1), whichimplies h(1) = 1. Taking the derivative of both sides and evaluating at 1 we get

0 = 2f(1)f ′(1) =

(d

dz

zN − 1

z − 1h(z)

)∣∣∣∣z=1

=

(d

dz

(1 + (z − 1))N − 1

z − 1h(z)

)∣∣∣∣z=1

=

(∑

k

(N

k + 1

)(z − 1)kh′(z) + k

(N

k + 1

)(z − 1)k−1h(z)

)∣∣∣∣∣k=1

=

(Nh′(1) +

N(N − 1)

2

)h(1)

= h(1) − h′(1),

and hence h′(1) = 1. Solving for h(z) we get h(z) = z, which implies

f(z)2 =zN+1 − 1

z − 1.

Let N + 1 = Mpα where (M, p) = 1. We will first prove that M = 1. Note thatsince we are working in characteristic p we get

zN+1 − 1 = zMpα − 1 = (zM − 1)pα

.


However we have

zN+1 − 1

z − 1= (z − 1)pα−1(1 + z + · · · + zM−1)pα

is a perfect square. However, 1 + z + · · · + zM−1 is square free and hence (1 + z +· · · + zM−1)pα

is not a perfect square, unless M = 1. This proves that N + 1 = q,

and f(z)2 = (z − 1)N . This give us f(z) = ±(z − 1)N/2. Hence

ai = −f(ri) = ±(ri − 1)N/2 = ±χ(ri − 1)

and hence

f(z) = ±∑

χ(ri − 1)zi

as desired. �

Remark 4.2. Note that the only place where we used the assumption that f(r) =

f(r2) = · · · = f(rN/2−1) = 0 was to bound the degree of f , or equivalently to show

that h is linear. This in turn is sufficient to find exact value for f(z)2. Without

this assumption we can find f(z)2 modulo zN − 1, however we do not know how touse this to show N + 1 = q.

5. Computational Verification of Conjectures & Final Comments

We have done extensive computations on the space of Littlewood-like polynomi-als in support of some of our conjectures.

First thought, it is worth observing a simple

Fact 5.1. Let f ∈ LN , and ζ be a primitive N th root of unity.

• If g(z) = ±f(±z) then

maxi

|f(ζi)| = maxi

|g(ζi)|

• Let gcd(k, N) = 1 and g(x) ≡ f(xk) (mod xN − 1), where g(x) ∈ LN .

Then

maxi

|f(ζi)| = maxi

|g(ζi)|

The proof is left to the interested reader.For the data collected for the conjecture, we

(1) Constructed all Fekete-like polynomials as given in Theorem 3.1 up to de-gree N = 500.

(2) Found all optimal polynomial f ∈ LN , with N ≤ 42 and all f ∈ AN , withN ≤ 84 such that maxi |f(ζi)| =

√N + 1

There were 16618 Fekete-like polynomials found using Theorem 3.1 up to degreeN = 500.

There were 700 optimal polynomials found through explicit search, as describedin (2) above. The first observation in Fact 5.1 was explicitly used in this secondcomputations. We only searched for those f ∈ LN , or AN where f(x) = x+x2±. . . .This cut our search space by a factor of 4.

Some data is provide in Table 1 and Table 2, for small degree examples. The firstobservation in Fact 5.1 was used, and only those examples were f(x) = x+x2± . . .are given. The full data can be found at [11].


Of the 700 optimal polynomial found by Test 2, almost all satisfied f(±1)2 = 1.The only exception we found was when N = 8, and is given by the polynomial

f8(z) = −x7 + x6 − x5 + x4 + x3 + x2 + x

along with the 3 related polynomials given by Fact 5.1. The authors conjecturethat these are the only exceptions.

As well, in Theorem 4.1, we assumed that there exists an r, a primitive Nthroot of unity in Fq, such that f(r) = f(r2) = · · · = f(rN/2−1) = 0 in Fq. There aretwo known exception to the case which are still optimal polynomial. The first is f8

above. The second is when N = 58 and is given by

f58 = x57 − x56 − x55 + x54 + x53 + x52 + x51 + x50 + x49 + x48 − x47

+x46 + x45 − x44 − x43 − x42 − x41 + x40 − x39 − x38 − x37 + x36

+x35 − x34 + x33 + x32 − x31 + x30 − x29 − x28 − x27 − x26 + x25

+x24 + x23 − x22 − x21 + x20 − x19 − x18 − x17 + x16 − x15 + x14

+x13 − x12 − x11 − x10 + x9 − x8 + x7 − x6 + x5 − x4 − x3 + x2 + x

along with its 7 related polynomials given by Fact 5.1. Here the roots seem tosatisfy a much different property. Namely, if f(r) ≡ 0 (mod 59) then f(r7) ≡ 0(mod 59). As {7i} (mod 58) has an orbit of size 7, this partitions the roots of f58

into 4 groups of size 7, 14 of which are primitive, and 14 of which are not.With the exception of f8 and f58 above, and their related polynomials, of the

polynomials found, if f is an optimal polynomial then f is a Fekete-like polynomial,as constructed by Theorem 3.1.

In our search for examples of polynomials, we restricted our search to f ∈ AN

when N ≥ 44. When N ≤ 42, we searched all f ∈ LN . In this search for N ≤ 42,which found 216 polynomials, we did not find any polynomial in LN \ AN . Weconjecture that all optimal polynomials are anti-skewsymmetric.

In the proof of Theorem 1.3, we needed f ∈ AN to give a lower bound on∑

c2k.

If we allow f ∈ LN , we could get all ck ≡ 0 (mod 4) for k 6= 0, (for example, usef = x + x2 + · · · + x5), so we cannot use a modular argument to get the desiredinequality. It is not clear what methods should be used to replace this method.

It is worth observing that this problem is very reminiscent of a number of prob-lems that come up in the study of Barker polynomials. In these problems, one looksat a sequence of integers a0, a1, · · · , aN−1 ∈ {±1} and define the c′k =

∑i−j=k aiaj .

In this paper we are looking at the cyclic autocorrelation problem (summing overi − j ≡ k (mod N)), with a0 = 0, where as the Barker polynomial problem looksat the acyclic autocorrelation problem (summing over i− j = k) with a0 = ±1. Seefor example [6, 9, 10, 13].

In our proof of Theorem 3.1, we made use of Lemma 3.2. which states that:

∑

a∈Fq

χ(a)χ(a + b) =

{−1 if b 6= 0,

q − 1 if b = 0.(5.1)

In [7], it is shown that it is possible for this property (5.1) to hold, but for χ notto be a multiplicative function. In fact, they give an explicit example over F9 (seeExample 2 of [7]). The obvious method of computing

∑χ(ri−1)xi with these non-

multiplicative functions χ does not give us an optimal polynomial. That is partlybecause in the proof of Theorem 3.1 we also used the fact that χ is a multiplicative


function. It is not clear if these non-multiplicative functions are related to otheroptimal polynomials.

In the proof of Theorem 4.1, we needed to make an extra assumption on theroots of the optimal polynomial over some finite field. This assumption is clearlytoo restrictive since there exists other optimal polynomials that don’t satisfy thesaid assumptions. It is not clear if it would be possible to give a more completedescription than Theorem 3.1 such that we would have a complete description of alloptimal polynomials. Currently as it stands, we don’t even know that if equalityholds, then N + 1 must be a prime power, although all evidences seems to suggestthis.

One of the oldest conjectures, of Littlewood, concerning these types of problemsis, does there exists C1 and C2, and a family of polynomials h with ±1 coefficientssuch that

C1

√deg(h) ≤ |h(z)| ≤ C2

√deg(h)(5.2)

for all |z| = 1. For more discussion on this conjecture, see [3]. Proving the easierconjecture, with respect to the upper bound only, looked promising for the Feketepolynomials hN(z) =

∑(iN

)zi for an odd prime N , since maxi |hN (ζi)| =

√N .

Unfortunately the values off of these roots of unity is sufficiently large so that thepartial conjecture cannot be proved in this way. In particular, with the standardFekete polynomials, it is known that

2

π

√N log log(N) < max

|z|=1|hN (z)| ≪

√N log(N)(5.3)

See [12]. One obvious question from this is, what happens with the Fekete-like poly-nomials? Can bounds such as (5.3) be found? It clearly cannot satisfy Littlewood’soriginal conjecture, as f(±1)2 = 1 for all Fekete-like polynomials.

Computationally it appears that the maximum grows faster than√

N . In fact,it appears that for large N that

C1 log log(N)√

N < max|z|=1

|f(z)| < C2 log log(N)√

N

with C1 > 0.85 and C2 < 1.45. This has the same sort of order as the lower boundfor Fekete polynomials. See Figure 1 and 2.

One related questions to this concerns the L4 norm, or the Merit factor of suchpolynomials. In [4] an explicit value for the L4 norm, and Merit factors for Feketepolynomials is given. Recall the Lα norm and Merit factor are defined as

||f ||α =

(1

2π

∫ 2π

0

|f(eiθ|α dθ

)1/α

MF(f) =||f ||42

||f ||44 − ||f ||42Here the L2 norm of a polynomials is the square root of the sum of the squaresof the coefficients, so in our case

√N − 1. For example, they show that, for hN a

Fekete polynomial that

||hN ||44 =5

3N2 − 3N +

4

3− 12C(−q)2

where C(−q) is the class number of Q(√−q). The expected value of the L4 norm

of a Littlewood polynomial is 21/4√

N , which gives a Merit factor of 1. For Fekete


SupNorm, C*ln(ln(x))*sqrt(x), C = 0.85, 1.1, 1.45

0

10

20

30

40

50

60

100 200 300 400 500

x

Figure 1. Sup Norm of Fekete-like polynomials

SupNorm/ln(ln(x))/sqrt(x)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

100 200 300 400 500

Figure 2. Sup Norm of Fekete-like polynomials – Normalized by log(N)√

N


Merit Factor, line 3.0

2

4

6

8

10

12

100 200 300 400 500

x

Figure 3. Merit Factor of Fekete-like polynomials

polynomials, the asymptotic Merit factor is 3/2, and if we move to the Turynpolynomials (which are a cyclic shift of the Fekete polynomials) we can get upto a Merit factor of 6. An obvious question is, what happens with Fekete-likepolynomials? It appears that the L4 norm grows like C

√N where 1.04 < C < 1.11.

Computationally it appears that that the Merit factors of these polynomials istending to 3 for large N . See Figure 3, 4 and 5

Another property of Fekete polynomials that has been much studied is the lo-cations of their roots. Initially, if we only look at primes less than 1000, there arevery few (23 in total) Fekete polynomials that have real roots in the interval (0, 1).This trend does not continue. In particular, Baker and Montgomery [1] show thatfor almost all large primes N , that the Fekete polynomial hN has a large numberof zeros in this interval. See also [2] for an alternate discussion on this topic. In [8],it is shown that more than half of the roots of Fekete polynomials are on the unitcircle. An obvious question is again, what happens with Fekete-like polynomials?Computationally it appears that the Fekete-like polynomials have no roots on theunit circle, ever. The Fekete-like polynomials can have roots in the interval (0, 1),and appears to happen quite often (about 3/4 of the time). In the other direc-tion, we haven’t found any Fekete-like polynomial with more than 5 real root inthis interval. Based on the data, it appears likely that the number of roots growssomewhat with N , so it should be possible to find polynomials with an arbitrarynumber of roots in [0, 1], for large enough N . See Figure 6


L_4 Norm, C*sqrt(x), C = 1.04, 1.08, 1.11

5

10

15

20

25

100 200 300 400 500

x

Figure 4. L4 norm of Fekete-like polynomials

L_4 Norm/sqrt(x)

1.02

1.04

1.06

1.08

1.1

1.12

1.14

1.16

0 100 200 300 400 500

Figure 5. L4 norm of Fekete-like polynomials – normalized by√

N

FE

KE

TE

-LIK

EP

OLY

NO

MIA

LS

15

N Polynomial max |f(ζi)| Sup Norm L4 norm Merit Factor # Real Roots

4 ++-

√5 2.2361 1.8212 4.5000 0

6 ++--+

√7 3.6056 2.5900 1.2500 0

8 +++--+-

√9 3.1034 2.7233 8.1667 0

10 ++-++---+

√11 5. 3.3166 2.0250 0

10 ++---+--+

√11 5.0000 3.3166 2.0250 1

12 +++---+--+-

√13 3.8023 3.3831 12.100 1

12 ++----+-++-

√13 5.2086 3.7369 1.6351 1

16 +++-+--++----+-

√17 5.4574 4.1722 2.8846 1

16 ++-----++-+-++-

√17 5.2957 4.1722 2.8846 1

16 ++++--+---++-+-

√17 5.5205 4.1722 2.8846 0

16 ++-++-+----+++-

√17 5.2957 4.1722 2.8846 0

18 ++++---++--+--+-+

√19 6.0470 4.3829 3.6125 0

18 ++-+-++--++-----+

√19 6.0520 4.3829 3.6125 1

18 ++--+----+-+++--+

√19 8.0623 4.7216 1.3894 1

22 +++-----++--++-+-++-+

√23 6.7082 4.8228 4.4100 0

22 ++-++++--+---++-+---+

√23 6.7082 4.8228 4.4100 0

22 ++----+---++-+++-+--+

√23 7.3351 5.0858 1.9342 1

22 +++----+--++---+-++-+

√23 7.3047 5.0858 1.9342 1

22 ++++--+--+---+++--+-+

√23 7.6593 5.0858 1.9342 0

24 +++-+-++-++--+++-----+-

√25 7.6006 5.2209 2.4720 0

24 ++--+-+++++--+-+----++-

√25 7.9767 5.2209 2.4720 0

26 +++--+--+----+-+++---++-+

√27 8.5440 5.6248 1.6622 1

26 ++++-+--++-++---++----+-+

√27 8.2799 5.4358 2.5202 0

28 +++--++-+-++++-+-----++--+-

√29 8.5230 5.7260 2.1069 0

28 +++-+++---++++-+--+--+---+-

√29 7.3451 5.4512 4.7338 0

28 ++-+--+----+--+++-+---++++-

√29 7.7758 5.5474 3.3440 1

28 +++++---+-++--++----+--+-+-

√29 7.3457 5.4512 4.7338 1

28 ++++-----+--++--+++-+-++-+-

√29 7.6504 5.5474 3.3440 0

28 ++-+-----++-++---++-+-++++-

√29 8.9304 5.6388 2.5851 2

30 ++-+-+--++-++-+++---++------+

√31 9.2195 5.7708 3.1381 1

30 ++-++-+++--+-+-----++-+++---+

√31 9.2195 5.7708 3.1381 0

30 +++-++-----+++--+--+-+--+++-+

√31 9.2195 5.9305 2.1237 0

30 +++--+++------++-+-+--+--++-+

√31 7.2801 5.5966 6.0071 1

Table 1. Small Fekete-like polynomials


Number of real roots in (0,1)

–1

0

1

2

3

4

5

100 200 300 400 500

Figure 6. Number of Real Roots of Fekete-like polynomials

References

[1] R. C. Baker and H. L. Montgomery, Oscillations of quadratic L-functions, Analytic numbertheory (Allerton Park, IL, 1989), Progr. Math., vol. 85, Birkhauser Boston, Boston, MA,1990, pp. 23–40.

[2] Paul T. Bateman, George B. Purdy, and Samuel S. Wagstaff, Jr., Some numerical results

on Fekete polynomials, Math. Comput. 29 (1975), 7–23, Collection of articles dedicated toDerrick Henry Lehmer on the occasion of his seventieth birthday.

[3] Peter Borwein, Some old problems on polynomials with integer coefficients, Approximationtheory IX, Vol. I. (Nashville, TN, 1998), Vanderbilt Univ. Press, Nashville, TN, 1998, pp. 31–50.

[4] Peter Borwein and Kwok-Kwong Stephen Choi, Explicit merit factor formulae for Fekete

and Turyn polynomials, Trans. Amer. Math. Soc. 354 (2002), no. 1, 219–234 (electronic).[5] Peter Borwein, Kwok-Kwong Stephen Choi, and Soroosh Yazdani, An extremal property of

Fekete polynomials, Proc. Amer. Math. Soc. 129 (2001), no. 1, 19–27 (electronic).[6] Peter Borwein, Erich Kaltofen, and Michael J. Mossinghoff, Irreducible polynomials and

Barker sequences, ACM Commun. Comput. Algebra 41 (2007), no. 3-4, 118–121.[7] Kwok-Kwong Choi and Man-Keung Siu, Counter-examples to a problem of Cohn on classi-

fying characters, J. Number Theory 84 (2000), no. 1, 40–48.[8] B. Conrey, A. Granville, B. Poonen, and K. Soundararajan, Zeros of Fekete polynomials,

Ann. Inst. Fourier (Grenoble) 50 (2000), no. 3, 865–889.[9] M. Elia, On the nonexistence of Barker sequences, Combinatorica 6 (1986), no. 3, 275–278.

[10] Jonathan Jedwab and Sheelagh Lloyd, A note on the nonexistence of Barker sequences, Des.Codes Cryptogr. 2 (1992), no. 1, 93–97.

[11] K. G. Hare, Home page, http://www.math.uwaterloo.ca/∼kghare, 2010.[12] Hugh L. Montgomery, An exponential polynomial formed with the Legendre symbol, Acta

Arith. 37 (1980), 375–380.[13] R. Turyn and J. Storer, On binary sequences, Proc. Amer. Math. Soc. 12 (1961), 394–399.


N Polynomial max |f(ζi)|22 + 1.0000004 ++- 5.0000006 ++--+ 7.0000008 ++++-+- 9.0000008 +++--+- 9.00000010 ++-++---+ 11.00000010 ++---+--+ 11.00000012 +++---+--+- 13.00000012 ++----+-++- 13.00000014 +++++-+--++-- 19.31766714 +++-+++-+--+- 19.31766714 +++---+-++-++ 19.31766714 +++----+---+- 19.31766714 ++-++-+---+++ 19.31766714 ++--++-+----- 19.31766716 ++++--+---++-+- 17.00000016 +++-+--++----+- 17.00000016 ++-++-+----+++- 17.00000016 ++-----++-+-++- 17.00000018 ++++---++--+--+-+ 19.00000018 ++-+-++--++-----+ 19.00000018 ++--+----+-+++--+ 19.00000020 +++++-+-++---++--+- 25.00000020 +++++-+-+--+--++--- 25.00000020 ++++--+-++---+---+- 25.00000020 ++++--+-+--+---+--- 25.00000020 +++-+++-++-+-++---- 25.00000020 +++-+++-++----++-+- 25.00000020 +++--++-++-+-+----- 25.00000020 +++--++-++-----+-+- 25.00000022 ++++--+--+---+++--+-+ 23.00000022 +++----+--++---+-++-+ 23.00000022 +++-----++--++-+-++-+ 23.00000022 ++-++++--+---++-+---+ 23.00000022 ++----+---++-+++-+--+ 23.000000

Table 2. Small Littlewood-like

Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada,

N2L 3G1

E-mail address: [email protected]

Department of Mathematics, McMaster University, Hamilton, Ontario, Canada, L8S

4L8

E-mail address: [email protected]

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